The rank of a signed graph in terms of the rank of its underlying graph

Let Γ=(G,σ) be a signed graph and A(Γ) be its adjacency matrix, where G is the underlying graph of Γ. The rank r(Γ) of Γ is the rank of A(Γ). We know that for a signed graph Γ=(G,σ), Γ is balanced if and only if Γ=(G,σ)∼(G,+). That is, when Γ is balanced, then r(Γ)=r(G), where r(G) is the rank of the underlying graph G of Γ. A natural problem is that: how about the relations between the rank of an unbalanced signed graph and the rank of its underlying graph? In this paper, we first prove that r(G)−2d(G)≤r(Γ)≤r(G)+2d(G) for an unbalanced signed graph with d(G)≥1, where d(G)=|E(G)|−|V(G)|+ω(G) is the dimension of cycle spaces of G, ω(G) is the number of connected components of G. As an application, we also prove that 1−d(G)<[Formula presented]≤1+d(G) for an unbalanced signed graph with d(G)≥1. All corresponding extremal graphs are characterized.